An Adaptive Controller Design for Flexible-joint Electrically-driven Robots With Consideration of Time-Varying Uncertainties 91 Appendix Lemma A.1: Let s ∈ ℜn , ε ∈ ℜn and K is the n × n positive definite matrix Then, ε − s Ks + s ε ≤ [λmin (K ) s − ] λmin (K ) T T (A.1) Proof: − s T Ks + s T ε ≤ [−λ (K ) s + ε ] s ε = − [ λ (K ) s − ]2 λmin (K ) ε − [λ (K ) s − ] λmin (K ) ε ≤ − [λ (K ) s − ] λmin (K ) Q.E.D Lemma A.2: Let w T = [ wi1 i defined as wi L win ] ∈ ℜ1×n , i=1,…,m and W is a block diagonal matrix W = diag{w1 , w ,L, w m } ∈ ℜmn×m Then, m Tr ( W T W ) = ∑ w i i =1 The notation Tr(.) denotes the trace operation Proof: The proof is straightforward as below: (A.2) 92 Frontiers in Adaptive Control ⎡ w11 ⎢0 WT W = ⎢ ⎢ M ⎢ ⎣0 L w1n L L w21 L w2 n L L M O O M M O M O L ⎡w ⎢ =⎢ ⎢ M ⎢ ⎣ 0 L L wm1 L ⎤ ⎡ w1 ⎥⎢ L ⎥⎢ O M ⎥⎢ M ⎥⎢ L wT ⎦⎣ m w ⎡ w w1 ⎢ =⎢ ⎢ M ⎢ ⎣ L T M T w1 L L T ⎡ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣ ⎡ w11 ⎢ M ⎢ ⎢ w1n ⎢ ⎤⎢ 0 ⎥⎢ M ⎥⎢ M ⎥⎢ ⎥ wmn ⎦ ⎢ M ⎢ ⎢ ⎢ M ⎢ ⎢ ⎣ M 0 w2 M 0 L M O L w21 L M O w2 n L M O L M O L ⎤ M ⎥ ⎥ ⎥ ⎥ ⎥ M ⎥ ⎥ ⎥ M ⎥ ⎥ wm1 ⎥ M ⎥ ⎥ wmn ⎥ ⎦ ⎤ L ⎥ ⎥ O M ⎥ ⎥ L wm ⎦ L ⎤ ⎥ ⎥ O M ⎥ ⎥ L wT w m ⎦ m L w w2 L T w2 M L M L O L 0 ⎤ ⎥ ⎥ M ⎥ 2⎥ wm ⎥ ⎦ The last equality holds because by definition w T w i = wi21 + wi22 + + wim = w i i m Tr = ( W T W) = ∑ w i Therefore, we have Q.E.D i =1 Lemma A.3: Suppose i=1,…,m w T = [ wi1 i Let W wi L win ] ∈ ℜ1×n and V be W = diag{w1 , w ,L, w m } ∈ ℜ respectively Then, block mn×m and diagonal and v T = [vi1 vi L vin ] ∈ ℜ1× n , i matrices that are defined V = diag{v1 , v ,L, v m } ∈ ℜ as mn × m , An Adaptive Controller Design for Flexible-joint Electrically-driven Robots With Consideration of Time-Varying Uncertainties 93 m Tr (V T W) ≤ ∑ v i w i (A.3) i =1 Proof: The proof is also straightforward: T ⎡ v1 ⎢ VT W = ⎢ ⎢M ⎢ ⎣0 L ⎤ ⎡ w1 ⎥ L ⎥⎢ ⎢ O M ⎥⎢ M ⎥⎢ L vT ⎦ ⎣ m vT M T ⎡ v1 w ⎢ =⎢ ⎢ M ⎢ ⎣ 0 v w2 M T w2 M L ⎤ L ⎥ ⎥ O M ⎥ ⎥ L wm ⎦ ⎤ L ⎥ ⎥ L O M ⎥ ⎥ L vT w m ⎦ m Hence, T Tr (V T W ) = v w + v T w + + v T w m m ≤ v w + v w + + v m w m Q.E.D m = ∑ vi wi i =1 Lemma A.4: Let W be defined as in Lemma A.2, and ˆ W ~ W is a matrix defined as ~ ˆ W = W − W , where is a matrix with proper dimension Then 1 ~ ˆ ~ ~ Tr ( W T W ) ≤ Tr ( W T W) − Tr ( W T W) 2 Proof: (A.4) 94 Frontiers in Adaptive Control ~ ˆ ~ ~ ~ Tr ( W T W ) = Tr ( W T W) − Tr ( W T W ) m ~ ~ ≤ ∑( wi wi − wi ) (by Lemma A.2 and A.3) i =1 = m ∑[ w i i =1 ~ − wi ~ − ( wi − wi )2 ] m ~ ∑( wi − wi ) i =1 1 ~ ~ = Tr ( W T W) − Tr ( W T W) (by Lemma A.2) 2 ≤ Q.E.D In the above lemmas, we consider properties of a block diagonal matrix In the following, we would like to extend the analysis to a class of more general matrices Lemma A.5: Let W be a matrix in the form W T = [ W1T Wi = diag{w i1 , w i ,L, w im } ∈ ℜmn ×m , entries of vectors T ij w = [ wij1 T L Wp ] ∈ ℜ pmn × m W2T where i=1,…,p, are block diagonal matrices with the wij L wijn ] ∈ ℜ1× n , j=1,…,m Then, we may have p m Tr ( W T W ) = ∑∑ w ij i =1 j =1 Proof: ⎡ W1 ⎤ ⎥ T ⎢ W T W = [ W1T L Wp ]⎢ M ⎥ ⎢ Wp ⎥ ⎣ ⎦ T T = W1 W1 + L + Wp Wp Hence, we may calculate the trace as (A.5) An Adaptive Controller Design for Flexible-joint Electrically-driven Robots With Consideration of Time-Varying Uncertainties 95 T Tr ( W T W ) = Tr ( W1T W1 ) + L + Tr ( W p W p ) m = ∑ w1 j j =1 p m + L + ∑ w pj m = ∑∑ w ij (by Lemma A.1) j =1 i =1 j =1 Q.E.D Lemma A.6: p Let V and W be matrices defined in Lemma A.5, Then, m Tr (V T W) ≤ ∑∑ v ij w ij i =1 j =1 (A.6) Proof: T Tr (V T W) = Tr (V1T W1 ) + L + Tr (V p W p ) m m j =1 j =1 ≤ ∑ v j w j + L + ∑ v pj w pj p (by Lemma A.3) m = ∑∑ v ij w ij i =1 j =1 Q.E.D Lemma A.7: Let W be defined as in Lemma A.5, and ˆ W ~ W is a matrix defined as ~ ˆ W = W − W , where is a matrix with proper dimension Then 1 ~ ˆ ~ ~ Tr ( W T W ) ≤ Tr ( W T W) − Tr ( W T W) 2 (A.7) 96 Frontiers in Adaptive Control Proof: ~ ˆ ~ ~ ~ Tr ( W T W ) = Tr ( W T W) − Tr ( W T W ) p m ~ ~ ≤ ∑∑ ( w ij w ij − w ij ) (by Lemma A.5 and A.6) i =1 j =1 = p m ∑∑ [ w ij i =1 j =1 ≤ p m ∑∑ ( w ij i =1 j =1 ~ − w ij ~ − w ij ) ~ − ( w ij − w ij ) ] 1 ~ ~ = Tr ( W T W) − Tr ( W T W) (by Lemma A.5) 2 Q.E.D Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems 1Chung-Yuan Chian-Song Chiu1,* and Kuang-Yow Lian2 Christian University, 2National Taipei University of Technology Taiwan, R.O.C Abstract This study proposes a novel adaptive control approach using a feedforward Takagi-Sugeno (TS) fuzzy approximator for a class of highly unknown multi-input multi-output (MIMO) nonlinear plants First of all, the design concept, namely, feedforward fuzzy approximator (FFA) based control, is introduced to compensate the unknown feedforward terms required during steady state via a forward TS fuzzy system which takes the desired commands as the input variables Different from the traditional fuzzy approximation approaches, this scheme allows easier implementation and drops the boundedness assumption on fuzzy universal approximation errors Furthermore, the controller is synthesized to assure either the disturbance attenuation or the attenuation of both disturbances and estimated fuzzy parameter errors or globally asymptotic stable tracking In addition, all the stability is guaranteed from a feasible gain solution of the derived linear matrix inequality (LMI) Meanwhile, the highly uncertain holonomic constrained systems are taken as applications with either guaranteed robust tracking performances or asymptotic stability in a global sense It is demonstrated that the proposed adaptive control is easily and straightforwardly extended to the robust TS FFA-based motion/force tracking controller Finally, two planar robots transporting a common object is taken as an application example to show the expected performance The comparison between the proposed and traditional adaptive fuzzy control schemes is also performed in numerical simulations Keywords: Adaptive control; Takagi-Sugeno (TS) fuzzy system; holonomic systems; motion/force control Introduction In recent years, plenty of adaptive fuzzy control methods (Wang & Mendel, 1992)-(Alata et al., 2001) have been proposed to deal with the control problem of poorly modeled plants All these researches are based on the fuzzy universal approximator (first proposed by Wang & Mendel, 1992), which is properly adjusted to compensate the uncertainties as close as possible Due to the use of states as the inputs of the fuzzy system, we call this approach as the state-feedback fuzzy approximator (SFA) based control In details, this methodology can be further classified into two types: i) Mamdani fuzzy approximator (Wang & Mendel, 1992; * Email: acs.chiu@gmail.com 98 Frontiers in Adaptive Control Chen et al., 1996; Lee & Tomizuka, 2000; Lin & Chen, 2002); and ii) Takagi-Sugeno (TS) fuzzy approximator (Ying, 1998; Tsay et al., 1999; Chen & Wong, 2000; Alata et al., 2001) The first type approach constructs the consequent part only via tunable fuzzy sets, but a good enough approximation usually requires a large number of fuzzy rules In contrast, the TS SFA-based controller uses the linear/nonlinear combination of states in consequent part such that fewer rules are required Without loss of generality, the configuration of these controllers is shown in Fig The SFA-based control contains the following disadvantages: i) numerous fuzzy rules and tuning parameters are required, especially for multivariable systems; ii) the fuzzy approximation error is assumed a priori to be upper bounded although the bound depends on state variables; and iii) the consequent part of TS fuzzy approximator will become complex for dealing with multivariable nonlinear systems, i.e., needing a complicated consequent part + + + − Figure Configuration of SFA-based adaptive controller + + + − Figure Configuration of FFA-based adaptive controller To remove the above limitations, this study introduces the feed-forward fuzzy approximator (FFA) based control which takes the desired commands as the premise variables of fuzzy rules and approximately compensates an unknown feed-forward term required during steady state (note that the configuration is illustrated in Fig 2) At the first glance, the SFA and FFA based control methods have a common adaptive learning concept, that is the feedback-error is used for tuning parameters of the compensator But, a closer investigation reveals the differences on: i) the type of training signals, ii) the process of taming dynamic uncertainties; and iii) the Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems 99 type of error feedback terms Especially, compared to SFA-based approaches (shown in Fig 1), the FFA-based adaptive controller needs a nonlinear damping term However, omitting feedback information in the fuzzy approximator leads to a less complex implementation (i.e., a simpler architecture compared to traditional SFA-based controllers) Furthermore, the fuzzy approximation error of FFA is always bounded, such that the synthesized controller assures global stability In addition, the number of fuzzy rules can be further reduced by using a TStype FFA In other words, the FFA-based adaptive controller has better advantages than the SFA-based adaptive controller To demonstrate the high application potential of the FFA-based adaptive control method to complicated and high-dimension systems, the FFA-based motion/force tracking controller is constructed for holonomic mechanical systems with an environmental constraint (McClamroch & Wang, 1988) or a set of closed kinematic chains (Tarn et al., 1987; Li et al., 1989) Holonomic systems represent numerous industrial plants — two for example, are constrained robots and cooperative multi-robot systems From the pioneering work (McClamroch & Wang, 1988), a reduced-state-based approach is utilized in most researches (Tarn et al., 1987; Li et al., 1989; Wang et al., 1997) When considering parametric uncertainties, adaptive control schemes were introduced in (Jean & Fu, 1993; Liu et al., 1997; Yu & Lloyd, 1997; Zhu & Schutter, 1999) Unfortunately, the reducedstate-based approach usually has a force tracking residual error proportional to estimated parameter errors Thus, a high gain force feedback or acceleration feedback is needed (e.g., Jean & Fu, 1993; Yu & Lloyd, 1997) An alternative hybrid motion/force control stated in (Yuan, 1997) has assured both motion and force tracking errors to be zero To deal with unstructured uncertainties, several robust control strategies (Chiu et al., 2004; Zhen & Goldenberg, 1996; Gueaieb et al., 2003) provide asymptotic motion tracking and an ultimate bounded force error In contrast to discontinuous control laws, the works (Chang & Chen, 2000; Lian et al., 2002) apply adaptive fuzzy control to compensate unmodeled uncertainties and achieve H ∞ tracking performance However, their applications are limited due to high computation load arising from the numerous fuzzy rules and tuning parameters All these points motivate the further research on improving the control of holonomic systems by using the FFA-based control As a result, the proposed adaptive controller is no longer with the disadvantages of the traditional SFA-based adaptive controllers mentioned above In detail, the stability is guaranteed in a rigorous analysis via Lyapunov’s method The attenuation of both disturbances and estimated fuzzy parameter errors is achieved in an L2 -gain sense, while the LMI techniques (Boyd et al., 1994) are used to simplify the gain design If applying the sliding mode control, the controlled system can further achieve asymptotic stability of tracking errors Notice that the proposed approach assures global stability for controlling general MIMO uncertain systems in a straightforward manner Compared to the mainly relative works (Chang & Chen, 2000; Lian et al., 2002), the proposed scheme achieves both robust motion and force tracking control (but the work (Lian et al., 2002) does not) for more general holonomic systems Meanwhile, the scheme has a novel architecture which can be easily implemented The remainder of this chapter is organized as follows First, the TS FFA-based adaptive control method is introduced in Sec Then, the proposed control method is modified to motion/force tracking controller for holonomic constrained systems in Sec Section shows the simulation results of controlling a cooperative multi-robot system transporting a common object Finally, some concluding remarks are made in Sec 100 Frontiers in Adaptive Control TS FFA-based Adaptive Fuzzy Control 3.1 FFA-based Compensation Concept Without loss of generality, let us consider an n-th order multivariable nonlinear system G( x(t ))x( n ) (t ) = f ( x(t )) + u(t ) + w(t ) (1) where n ≥ ; x ∈ Rm is a part of the state vector x defined as x(t ) = [ xT (t ) xT (t ) L & ( x( n−1) (t ))T ]T ∈ Rnm ; f ( x(t )) ∈ Rm is an unknown nonlinear function which satisfies f ( x d(t )) ∈ L∞ for an appropriate bounded desired tracking command x d(t ) = [ xT (t ) xT (t ) L &d d ( ( xdn−1) (t ))T ]T ; G( x(t )) ∈ Rm×m is an unknown positive-definite symmetric matrix which & satisfies G( x d(t )) , G( x d(t )) ∈ L∞ ; u(t ) ∈ Rm is the control input; and w(t ) ∈ Rm is an external disturbance assumed to be bounded Clearly, if the terms f ( x(t )) and G( x(t )) are exactly known and no disturbance exists, we are able to apply the feedback linearization concept and set the control law as & u = − f ( x ) + G( x )q a(t ) + & G( x )s + Ks (2) where the notations are given as e(t ) = xd (t ) − x(t ) , s(t ) = q a (t ) − x( n−1) (t ) , ( q a (t ) = xdn−1) (t ) & +Λ n−1 e( n−2 ) (t ) + L + Λ e(t ) + Λ e(t ) ; Λ v ∈ Rm×m , for v = 1, 2, , (n − 1) , is a positive-definite diagonal matrix; and K ∈ Rm×m is a symmetric positive-definite matrix This renders to the & & error dynamics G( x )s = − G( x )s − Ks − w(t ) , which is exponentially stable once there is no & & disturbance However, the state feedback term u = − f ( x ) + G( x )q (t ) + G( x )s is often poorly b a understood such that the fuzzy approximator is considered to realize the ideal control law (2) in conventional SFA-based control methods Nevertheless, when the tracking goal is achieved, terms f ( x(t )) and G( x(t )) accordingly converge to functions f ( x d(t )) and G( x d(t )) The state feedback term ub converges to u f = − f ( x d ) + G( x d )x(dn ) (3) which is only dependent on the pre-planned desired command x d In other words, the state feedback control law becomes a feedforward compensation law during steady state Therefore, different to traditional works (Wang & Mendel, 1992)-(Alata et al., 2001), here we use the universal fuzzy approximator to closely obtain the feed-forward compensation law (3), while the effect of omitting transient dynamics is compensated by error feedback Since the pre-planned desired commands would be taken as the inputs of the fuzzy approximator, the so-called feed-forward fuzzy approximator (FFA) arises By this way, we assume that there exist positive constants ψ , ,ψ p and positive-semidefinite symmetric matrices Ψ s , Ψ e such that the error between ub ( x ) and u f ( x d ) is shaped by p sT (ub ( x ) − u f ( x d )) ≤ ∑ψ κ =1 κ eo 2κ T s + sT Ψ s s + e o Ψ e e o (4) Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems 101 with the tracking error eo = x − x d Then the design idea can be realized by combining both FFA and error-feedback based compensations later Note that the above inequality is often held for most physical systems, such as robotic systems, dc motors, etc Moreover, p = is often held The similar property as (4) for nonlinear systems can be found in (Sadegh & Horowitz, 1990; Chiu et al., 2006; Chiu, 2006) From the definition of u f in (3), the TS-type FFA consists of the following rules: Rule l : If z1 (t ) is X l1 and and zh (t ) is X lh Then l l ˆ u fi = θ i + θ 1i χ ( x d ), l = 1, 2, , r (5) where z1 (t ) , , zh (t ) are the premise variables composed of the desired commands xd (t ) , ( n−1) & x d(t ) , , xd (t ) since u f ( x d(t )) is functional of x d(t ) ; l = 1, 2, , r with r denoting the total number of rules; X l1 , , X lh are proper fuzzy sets determined by the known behavior ˆ of the desired signals; u fi is the i-th element of approximation of u f ; χ ∈ R g is a basis vector functional of x d(t ) to be chosen from the nonlinearity of u f ; θ 0l i ∈ R and θ 1l i ∈ R 1×g are fuzzy parameters Using the singleton fuzzifier, product fuzzy inference and weighted average defuzzifier, the inferred output of the fuzzy system (5) is T ˆ u fi( zd ,Θu fi ) = ξ ( zd )Θu fi χ ( zd ) where zd (t ) ≡ [ z1 (t ) z2 (t ) zh (t )]T ; Θu fi ≡ [θ u1fi (6) θ u2fi θ ur fi ]T ∈ R r×( g+1) with θ ul fi = [θ 0l i θ 1l i ]T ∈ R g+1 ; χ = [1 χ T ]T ∈ R g+1 ; and ξ ( zd (t )) ≡ [ξ1 ξ ξr ]T ∈ R r is a fuzzy basis function vector consisting of ξ l ( zd (t )) = μl ( zd (t ))/ ∑ r l =1 l μl ( zd (t )) with μl ( zd (t )) = ∏ ζ =1 Xζ ( zζ (t )) ≥ for h all l Note that the form of (6) is a TS type of fuzzy representation When we let χ = , the fuzzy system (5) is reduced to the special case with a Mamdani fuzzy representation, i.e., T r ˆ u fi = ξ ( zd )Θu fi for Θu fi ∈ R and χ = Based on the above fuzzy approximator (6), the overall approximation of u f is obtained as ⎡ ⎤ ˆ ˆ u f ( zd (t ),Θu f ) = ⎢⎢⎢⎣ u fi( zd (t ),Θu fi )⎥⎥⎥⎦ m×1 = Yd ( zd (t ))Θu f χ (7) where Θu f = [ΘT f ΘT f L ΘT fm ]T ∈ R mr×( g+1) ; and Yd = block-diag {ξ T , , ξ T } ∈ R m×mr is a u u u regression matrix From the observation on (7), if Θu f is bounded, then u f ∈ L∞ for all t ˆ (due to Yd ( zd (t )) ∈ L∞ and χ ∈ L∞ for all bounded x d(t ) ) In light of this, we limit the tunable fuzzy parameter Θu f to a specified region { } Ωθu ≡ Θu f ∈ R mr×( g+1) tr(ΘT f Θu f ) ≤ θ u, θ u > u with an adjustable parameter θ u Meanwhile, an appropriate projection algorithm will be applied later to keep the tuned fuzzy parameters within the bounded region Inside the 102 Frontiers in Adaptive Control specified set, there exists an optimal approximation parameter Θ∗ f defined as (for U z is a u discussed space of zd ) Θ∗ f ≡ argmin Θu u f ∈Ωθu ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ sup zd∈Uz u f − u f ( zd ,Θu f ) ⎟⎟⎟⎟ ˆ ⎠ which leads to the minimum approximation error for u f This means that the minimum approximation error is Wu f = u f ( zd ) − Yd ( zd (t ))Θ∗ f χ u (8) Note that if the parametric constraint is removed, the optimal approximation parameter Θ∗ f u ˆ is still upper bounded (cf Wang & Mendel, 1992) Due to u f ( zd ,Θ∗ f ) ∈ L∞ and u f ( x d ) ∈ L∞ , it u is reasonably concluded that Wu f is upper bounded for all t Moreover, based on the universal approximation theorem (Wang & Mendel, 1992), Wu f can be arbitrarily small In addition, special characteristics of the feedforward fuzzy approximator are summarized below Next, according to the FFA (7) and the bounded fashion of ub ( x ) − u f ( x d ) as (4), the overall controller with an adaptively tuned FFA is given as follows: p u = u f ( zd ,Θu f ) + ˆ ∑ψ κ =1 κ eo 2κ s + Ks (9) ⎧ tr( χ sT Yd Θu f ) Θu f , if (cu (Θu f ) ≥ and ⎪γ 0YdT s χ T − γ 0c u (Θu f ) tr(ΘT f Θu f ) ⎪ u ⎪ & tr( χ sT Yd Θu f ) > 0) Θu f = ⎨ ⎪ T γ Ed s χ T , otherwise ⎪ ⎪ ⎩ (10) where γ > ; cu (Θu f ) = ( tr (ΘT f Θu f ) − θ u + ε u )/ε u with cu (Θu f (t0 )) < and θ u > ε u > Note u that the above update law is an application of the smooth projection algorithm developed in the work (Pomet & Praly, 1992) The update law assures the following properties: (a) &u % % % tr (ΘT f Θu f ) ≤ θ u for all t ≥ t0 and (b) γ tr( χ sTYd Θu f ) − tr (ΘT f Θu f ) ≤ for Θu f = Θ∗ f − Θu f u u Then, the controller (9) results in the overall error system p & & G( x )s = − G( x )s − ψ κ eo κ =1 ∑ & e= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2κ % s − Ks + Yd Θ u f χ + Δu + wa (t ) Im L e M O O M M L I −Λ L −Λ n−2 ⎤⎡ ⎤ ⎥⎢ ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ ⎥ ⎢ ( n−3) ⎥ ⎥ m ⎥⎢ ⎥⎢ ⎥ ⎥ ⎢ ( n− ) ⎥ ⎥ n −1 ⎥ ⎢ ⎦⎣ ⎦ −Λ e e + ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ m⎥ ⎣ ⎦ M I s (11) Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems ≡ Λ e + Bs 103 (12) % where Θu f = Θ∗ f − Θu f ; wa (t ) = Wu f − w(t ) ; Δu = ub ( x ) − u f ( x d ) ; the definition of Wu f as (8) u ( n− ) T T ) ] ∈ R m( n−1) ; Λ ∈ R m( n−1)×m( n−1) and B ∈ R m( n−1)×m are & eT L ( e defined from the above associated components Since the error system (11) is only perturbed by the bounded approximation error wa (t ) , the globally uniform ultimate bound of eo is has been used; e = [ eT assured straightforwardly The detailed stability analysis will be carried out in the next subsection 3.2 Robustness Design To further enhance the robustness of the controlled system, three modified FFA-based adaptive controllers are developed in this subsection First, the robust gain design is performed here Let us consider the Lyapunov function candidate V1 (t ) = T %u % tr(ΘT f Θ u f ) s G( x )s + eTPe + 2γ (13) with a positive-definite symmetric matrix P The time derivative of V along the error dynamics (11) and (12) is T T T T T T T T & V = −s Ks + e ( Λ P + PΛ )e + s B Pe + e PBs + s Δu + s wa p 2κ &u % % − ψ κ eo sT s + tr sT Yd Θu f χ − tr(ΘT f Θ u f ) ( ∑ κ =1 T ) γ0 ≤ −s (K − Ψ s )s + e ( Λ P + PΛ )e + s BT Pe + eTPBs T T T + eT Ψ e eo + sT w a o &u % % % % where the facts tr(sTYd Θu f χ ) = tr( χ sTYd Θu f ) , tr(ΘT f Θu f ) ≥ γ tr(χ sT Yd Θu f ) and the inequality (4) have been e = [ I m( n−1) applied Furthermore, if & 0m( n−1)×m ]eo are applied, V satisfies the expressions ⎛⎡ ⎜⎢ s = [ −BT Λ I m ]eo and ⎞ ⎟ H − ΛT BK r BT Λ PB + ΛT BK r ⎤ ⎥ ⎥ + Ψ ⎟e + w a (t ) e⎟ o ⎥ T P+K BT Λ ⎟ B −K r ρ12 ⎥ ⎜⎢ ⎟ r ⎦ ⎝⎣ ⎠ T ⎜ & V ≤ eo ⎜ ⎢ ⎢ ⎜ where H = ΛT P + PΛ − ΛT BBT P − PBBT Λ and K r = K − Ψ s − ρ1 2 (14) Therefore, the robust control result is summarized in the following theorem Theorem 1: Consider the highly unknown system (1) using the TS FFA-based adaptive fuzzy controller (9) with the update law (10) If there exist symmetric positive-definite matrices P , K satisfying the following LMI problem Given ρ > 0, Λ v > 0, Q ≥ subject to P, K > ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ H − ΛT BK r BT Λ PB + ΛT BK r ⎤ ⎥ ⎥ +Ψ +Q ≤ e ⎥ BT P + K r BT Λ −K r ⎥ ⎦ (15) 104 Frontiers in Adaptive Control then the closed-loop error system has the following properties: (i) all error signals and fuzzy parameters are bounded; (ii) the H ∞ tracking performance criterion ∫ tf t0 eT (t )Qeo (t )dt ≤ V1 (t0 ) + o ρ ∫ tf t0 wa (t ) 2dt (16) is assured; and (iii) if wa (t ) ∈ L2 , then eo asymptotically converges to zero in a global manner Proof: From the inequality (14), a feasible solution of the LMI (15) yields & V ≤ − eT Qeo + wa (t ) o (17) ρ Since { eo V1 > and & V1 is negative semidefinite outside the compact set % eo ≤ η0 ρ1 wa < ∞} , for η0 = λmin (Q )} , we have e, s ∈ L∞ and Θu f ∈ L∞ As a result, && e, s ∈ L∞ is assured from the boundedness of all terms on right-hand side of (11) and (12) In turn, eo , e o ∈ L∞ & Moreover, by integrating the inequality (17), the H ∞ tracking performance criterion (16) is assured In other words, the disturbance wa (t ) is attenuated to a prescribed level 1/ρ Also, eo ∈ L2 if wa (t ) is L2 integrable Due to the fact that eo , e o ∈ L∞ and eo ∈ L2 , the result & lim t→∞ eo (t ) = is concluded by Barbalat’s lemma In addition, since the augmented disturbance wa (t ) is naturally bounded, all the stability is in a global sense ▓ Furthermore, to avoid an unexpected transient response due to poor fuzzy approximation, the attenuation of fuzzy parameter errors is taken into consideration below Theorem 2: Consider the highly unknown system (1) using the TS FFA-based adaptive fuzzy controller u = u f ( zd ,Θu f ) + ( ˆ ∑ p ψ κ eo 2κ κ =1 + ρ 22 χ Y2 d )s + Ks (18) with ρ > , Y2 d = YdYdT = diag {ξ T ξ , , ξ T ξ } ∈ R m×m , and the update law (10) If there exist symmetric positive-definite matrices P , K satisfying the LMI problem (15), then the closedloop error system achieves the H ∞ tracking performance criterion ∫ tf t0 eT (t )Qeo (t )dt ≤ V2 (t0 ) + o ∫ tf t0 ( ρ12 wa (t ) + ρ 22 %u % tr(ΘT f (t )Θ u f (t )))dt (19) where V2 (t0 ) is a quadratic term dependent on the initial values of tracking errors; and % 1/ρ > is a prescribed attenuation level for the fuzzy parametric error Θu f Proof: Consider the Lyapunov function candidate V2 (t ) = T T s G( x )s + e Pe Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems 105 with symmetric positive-definite matrices G( x ) and P Similar to the proof in Thm 1, the feasibility of the LMI (15) and the control law (18) of u lead to T & V ≤ − eo Qeo + ρ wa 2 ( ) % + tr sTYd Θu f χ − ρ 22 χ Y2 d sT s From the property % tr(sT Yd Θu f χ ) ≤ ρ 22 tr( χ T χ sT Y2 d s ) + ρ 22 %u % tr(ΘT f Θu f ) & V further satisfies T & V ≤ − eo Qeo + ρ12 wa + ρ 22 %u % tr(ΘT f Θu f ) Integrating both sides of the above inequality, the closed-loop system guarantees the robust performance criterion (19) The gain ρ is the adjustable attenuation level of fuzzy parametric errors In addition, the boundedness of the error system is assured from the same argument in Thm ▓ From the observation on wa , the boundedness has been assured from the bounded fuzzy approximation output (7) and error (8) in a global sense This implies that there exists a conservative upper bound of wa to be a constant η such that η ≥ max{supt wai (t ) } (where i =1, ,m wai denotes the i -th element of the vector wa ) Then we are able to give an asymptotic stable result as below Theorem 3: Consider the highly unknown system (1) using the TS FFA-based adaptive fuzzy controller u = u f ( zd ,Θu f ) + ˆ ∑ p ψ κ eo κ =1 2κ s + Ks + η sign(s ) (20) with sign(s) ≡ [sign(s1 ) L sign( sp )] for si being the i -th element of vector s and the update law (10) If there exist symmetric positive-definite matrices P , K satisfying the following LMI problem (15) for given ρ = , then the tracking error asymptotically converges to zero in a global sense Proof: Consider the Lyapunov function candidate (13) again Analogous to the proof of Thm 1, the feasibility of the LMI (15) with ρ = and the control law (20) yield T T T & V ≤ − eo Qeo + s wa − η s sign(s ) ≤ − eT Qeo + o ∑ m i =1 si wai − η ∑ m i =1 si ≤− eTQeo o & where the upper boundedness of wa has been used Due to V1 > and V ≤ , we are able to conclude the tracking error e will asymptotically converge to zero as t → ∞ ▓ 106 Frontiers in Adaptive Control Remark 1: The proposed feedforward fuzzy system (5) has four important characteristics — (a) the premise variables only consist of desired commands such that some fuzzy inference steps (e.g., calculation of Yd ( zd (t )) ) can be performed off-line; (b) an assumption on the bounded approximation error is not needed; (c) due to the naturally bounded approximation error Wu f , the total number of fuzzy rules can be flexibly reduced if a large approximation error is acceptable; and (d) TS-type fuzzy rules provide more flexible approximation by using fewer rules Therefore, the feedforward fuzzy approximator allows less computation and the synthesized controller has simpler implementation along with a ▓ globally stable manner Application on Holonomic Systems 4.1 Model Descriptions of Holonomic Systems Consider a non-redundant holonomic system with a generalized coordinate q ∈ R m and the & holonomic constraint φ (q ) = and A(q )q = , where φ : R m a R p and A(q ) = ∂φ ( q ) ∂q Without loss of generality, we assume that the system is operated away from any singularity with the exactly known function φ (q ) ∈ C From investigation on well-known holonomic systems, different model descriptions exist due to the two kinds of constraints — an environmental constraint and a set of closed kinematic chains Nevertheless, the model’s general form is able to be formulated into a fully actuated system with a constraint Referring to (Chiu et al., 2006), the general model of a holonomic system is written as & & M(q )&& + C (q, q )q + g( q ) + τ d (t ) = Bgτ g + AT λg q (21) & & where M(q ) , C (q, q )q , g(q ) are the inertia matrix, Coriolis/centripetal force, gravitational force, respectively (which are continuous and assumed to be poorly known); τ d (t ) is a bounded external disturbance; τ g ∈ R m is an applied force; Bg (q ) is an invertible input matrix; and λg ∈ R p physically presents a reaction force for an environmental constraint or an internal force for a set of closed kinematic chains Since the motion is subject to a p -dimensional constraint, the configuration space of the holonomic system is left with (m − p ) degrees of freedom From the implicit function T theorem (McClamroch & Wang, 1988), we find a partition of q as q = [q1 T q2 ]T for q1 ∈ R m−p , q2 ∈ R p , such that the generalized coordinate q2 is expressed in terms of the independent coordinate q1 as q = Ω(q1 ) with a nonlinear mapping function Ω Due to the nonsingularity assumption, the terms ∂Ω ∂q1 and ∂ 2Ω ∂q1 are bounded in the work space The generalized displacement and velocity can be expressed in terms of the independent & coordinates q1 , q as T q = [ q1 ( Ω(q1 ))T ]T (22) 107 Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems ⎡ I n −m ⎤ & & & q = ⎢ ∂Ω(q1 ) ⎥ q ≡ J q ⎢ ⎥ ⎢ ∂q1 ⎥ ⎣ ⎦ (23) & & From above equations, the constraint of velocity A(q )q = leads to A(q1 ) J (q1 )q = Notice that here we use A(q1 ) to denote A(q1 ,Ω(q1 )) for brevity In other words, A(q1 ) J (q1 ) = & since A(q1 ) J (q1 ) is full column-rank and q is an independent coordinate (see (McClamroch & Wang, 1988)) Thus, there exists a reduced dynamics for the holonomic system (21) Due q && q to the velocity transformation (23), the generalized acceleration satisfies && = J &&1 + J q The & q motion equation (21) is further represented by the independent coordinates q1 , q 1, &&1 as && & & M (q1 ) J q + C (q1 , q )q + g(q1 ) + τ d (t ) = Bg ( q1 )τ g + AT λg (24) & where C = MJ + CJ According to the fact A(q1 ) J (q1 ) = , a reduced dynamics (McClamroch & Wang, 1988) is obtained after multiplying J T on both sides of (24): && & & M (q1 )q + C (q1 , q )q + g(q1 ) + τ d(q1 , t ) = J T Bgτ g (25) with M = J T MJ ; C = J T C ; g = J T g ; and τ d = J Tτ d From the dynamics (25), some useful properties are addressed below Property 1: For the partition T E2 = [0 p×( m−p ) I p ] ∈ R m×p I m = [E1 E2 ] with E1 = [ I m−p 0( m−p )×p ]T ∈ R m×( m−p ) and T , the velocity transformation matrix J satisfies J E1 = I m−p Property 2: From the existence of Ω(⋅) and the implicit function theorem, A2 is invertible −1 Property 3: The matrix M is symmetric and positive-definite while M ∈ L∞ Property 4: Matrix ( M − 2C ) is skew-symmetric (cf McClamroch & Wang, 1988), i.e., ζ T ( M − 2C )ζ = , ∀ζ ∈ R m−p 4.2 FFA-Based Adaptive Motion/Force Control For holonomic systems, the control objective is to track a desired motion trajectory q1 d (t ) ∈ C while maintaining force λg at a desired λgd (t ) Inspired by pure motion tracking, some notations are defined as em = q1 d − q1 , em ∈ R m−p ; & q a = Λ m em + q d , q a ∈ R m−p ; & s = q a − q , s ∈ R m− p ; (26) where em , q a , s are the motion error, auxiliary signal vector, error signal, respectively; and Λ m ∈ R( m−p )×( m−p ) is a symmetric positive-definite matrix If the system satisfies lim t→∞ s(t ) = , & then position and velocity tracking errors em , e m exponentially converge to zero In other 108 Frontiers in Adaptive Control words, the motion tracking problem is transformed to the problem of stabilizing s(t ) On the other hand, a force tracking error and force error filter are accordingly defined as % λ = λgd − λg ∈ R p (27) % & e λ + η1 eλ = η2 λ , with η1 ,η2 > (28) Then, the reduced-state based scheme is to drive the motion trajectory into the stable subspace while the contact force is separately controlled maintaining a zero eλ In order to derive the adaptive fuzzy controller, the error dynamics of s along the motion equation (24) is written as & & q MJs = MJ q a − MJ &&1 (29) = −Cs + f + τ d − AT λg − Bgτ g & & where f = M(q1 ) J (q1 )q a + C (q1 , q )q a + g(q1 ) ∈ R m By traditional SFA-based control, we & & q usually require to take q1 , q , q1 d , q d , &&1 d as the premise variables, such that a large computational load exists on the controller processor To avoid this situation, the FFA-based control method is used to provide the feed-forward compensation term & q & & q f d (q1 d , q d, &&1 d ) = M( q1 d ) J (q1 d )&&1 d + C (q1 d , q d )q d + g(q1 d ) Since f d is independent to state variables, f d (⋅) is a much simpler function than f (⋅) If the effect of omitting the error f − f d can be coped with by feedback of tracking error, the concept of using the forward compensation f d is feasible According to the FFA-based control in the above section, we closely approximate and compensate the forward term f d (⋅) by a TS fuzzy system with the singleton fuzzifier and product inference Then the fuzzy inferred output is ˆ f d( zd (t ),Θ fd ) = Yd ( zd (t ))Θ f d χ (30) where zd (t ) , Yd ( zd (t )) , and χ have the same definition as (7) being functional of & q q1 d (t ), q d(t ), &&1 d(t ) ; and Θ f d ∈ R mr×( g+1) is a fuzzy tuning parametric vector in the consequent part of rules, with r denoting the total number of rules For the FFA (30), there exists an optimal approximation parameter ⎛ ⎞ ⎝ ⎠ ˆ Θ∗f d ≡ argmin Θ fd ∈Ωθ fd ⎜⎜⎜ sup zd∈Uz f d − f d( zd (t ),Θ f d ) ⎟⎟⎟ in an appropriate parametric constraint region Ωθ fd , which provides the most accurate approximation with the minimum error: W f d = f d − Yd ( zd (t ))Θ∗f d χ (31) From the observation on the right-hand side of the above equation, the fuzzy approximation ˆ error W is upper bounded for t ≥ from f d ∈ L∞ and f ∈ L fd d ∞ Next, the overall controller is synthesized in the following Based on the TS FFA-based fuzzy system (30), the overall control law is set in the form: Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems − τ g = Bg [Yd Θ fd χ + E1 (Ks + τ a ) − AT (λgd + k f eλ )] 109 (32) where k f > is a force feedback gain; K ∈ R( m−p )×( m−p ) is a symmetric positive definite matrix; τ a is an auxiliary input designed later; the definition of s and eλ is given in (26) and (28), respectively Meanwhile, the fuzzy parameter Θ fd is adaptively adjusted by ⎧ T tr( χ sT J T Yd Θ fd ) T Θ fd , ⎪γ Yd Js χ − c(Θ f d ) tr(ΘTd Θ f d ) if (c( Θ f d ) ≥ and f ⎪ & fd = ⎪ tr( χ sT J TYd Θu f ) > 0) ⎨ Θ ⎪ γ YdT Js χ T , otherwise ⎪ ⎪ ⎩ with c(Θ fd (t0 )) < , where c(Θ f d ) = ( tr (ΘTd Θ f d ) − θ f fd (33) + ε f )/ε f is a projection criterion function with a tunable parameter ε f satisfying θ f d > ε f > ; and γ > is an adaptation gain Furthermore, substituting the control law (32) into the dynamic equation (29) renders the closed-loop error dynamics: % & % MJs = −Cs − E1 (Ks + τ a ) + Yd Θ f d χ + Δf + w + AT (λ + k f eλ ) (34) where Δf ≡ f − f d ; w ≡ W f d + τ d ; and the definition of approximation error W f d in (31) and % λ in (27) have been applied To analyze the convergence of motion and force tracking separately, we further multiply J T on both sides of (34), which leads to the motion tracking error dynamics: & % Ms = −Cs − Ks + J TYd Θ f d χ + J T Δf + w + τ a , (35) where Property ( J T E1 = I n−m ) and the fact, J T (q1 ) AT (q1 ) = 0, have been applied; and − T & w ≡ J T w Then, replacing s of (34) by (35) and multiplying A2 T E2 on both sides of (34), we obtain the force tracking error as follows: ( T − % % λ + k f eλ = A2 T E2 MJ M −1( −Cs − Ks + J T Yd Θ f d χ % + J Δf + w + τ a ) + Cs − Δf − Yd Θ fd χ − w T % ≡ ϖ ( em , s , Θ f d , w , t ) ) (36) − T where Property ( A2 T ∈ L∞ ) and the fact, E2 E1 = , have been applied above It is a worthwhile note that the perturbed term Δf in (35) arises from the use of the feed-forward fuzzy compensation Nevertheless, the term Δf is upper bounded by motion tracking errors in the following fashion: s T J T Δf ≤ s T ( Ψ s + 2κ 2 T I n−m )s + Λ m em sT Ψ se s + em ( Ψ e + κ2 Ψ J )em (37) 110 Frontiers in Adaptive Control where there exist an intermediate parameter κ > and symmetric positive semidefinite matrices Ψ s , Ψ se , Ψ e , Ψ J dependent on the desired motion trajectory, control parameter Λ m , and system parameters This boundedness is assured for all well-known holonomic mechanical systems (cf Appendix of (Chiu et al., 2006)) Now, the main results of the FFA-based adaptive control of holonomic systems are stated as follows Theorem 4: Consider the holonomic system (21) using the TS FFA-based adaptive controller (32) tuned by the update law (33) If the auxiliary input is set as τ a = Pm em + Λ m em Ψ ses + ρ2 χ T χ J T YdYdT Js (38) and there exist κ , K , Pm satisfying the following LMI problem ⎡ ⎢ Q11 ⎢ ⎣ Q21 Q22 ⎥ ⎥ ⎦ Given ρ , Λ m > and Q = ⎢ ⎢ T Q21 ⎤ ⎥ ⎥ >0 subject to κ , K , Pm > ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ with K a = K − ( ρ4 + T ΛT KT −Q21 κΨ1 / ⎤ m a J ⎥ ⎥ Ka −Q22 ⎥≥0 ⎥ ⎥ I n −m ⎥ ⎥ ⎦ K p − Q11 K a Λ m − Q21 κΨ / J (39) )I n−m − Ψ s and K p = ΛT K a Λ m + ΛT Pm + Pm Λ m − Ψ e , then (a) error signals m m 2κ % % em , e m , eλ , λ and fuzzy parameter Θ f d are bounded; (b) error vectors em , s , λ have & globally uniform ultimate bounds being proportional to the inversion of control gains; and (c) the closed-loop system is guaranteed with the robust motion tracking performance ∫ tf t0 eT Qea dt ≤ Vs (t0 ) + a ρ2 ∫ tf t0 %f % ( w(t ) +tr(ΘTd (t )Θ f d (t )))dt (40) T for ea = [ em T &m eT ] and a nonnegative constant Vs (t0 ) Proof: First, we prove the claim (a) Consider the Lyapunov function candidate V= T T %f % tr ΘTd Θ fd s Ms + em Pm em + 2γ ( ) with a proper symmetric positive-definite matrix Pm Along the error dynamics (35) and the fact e m = −Λ m em + s , the time derivative of V is written as follows: & ρ2 T T T T & & V = sT ( M − 2C )s − sT Ks − em ( ΛT Pm + Pm Λ m )em − Λ m em sT Ψ ses − χ χ s J YdYdT Js m &f % % + sT J T Yd Θ fd χ − tr ΘTd Θ fd + sT J T Δf + sT w γ ( ) T ≤ −sT Ks − em ( ΛT Pm + Pm Λ m )em − Λ m em sT Ψ se s + sT J T Δf + sT w m 111 Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems where the definition of τ a , Property 4, and the update law (33) have been applied; and the above inequality is ensured by the property of the update law (i.e., &f % % sT J T Yd Θ fd χ − γ1 tr( ΘTd Θ f d ) ≤ ) Due to the boundedness of Δf as the fashion (37), we further obtain T & V ≤ −sT K a s − em ϒem + w (41) ρ where K a = K − ( ρ4 + 2κ the expressions s = [ Λ m )I n−m − Ψ s ; ϒ = ΛT Pm + Pm Λ m − Ψ e − κ2 Ψ J ; and ρ > Then, applying m In−m ]ea and em = [ I n−m 0n−m ]ea , the inequality (41) is rewritten as ⎛ ⎡ ΛT K Λ + ϒ ΛT K a ⎤ ⎞ ⎢ ⎥ m & ⎥ − Q⎟e V ≤ − eT ⎜ ⎢ m a m a ⎢ ⎥ ⎜⎢ ⎟ a K aΛ m Ka ⎥ ⎦ ⎝⎣ ⎠ − eTQea + w a ρ & Thus, if the LMI (39) has a feasible solution, then the following V holds & V ≤ −α ea + ρ2 w (42) & with α = λmin (Q ) Since V is positive-definite and V satisfies the inequality (42), we can & e % conclude that s, em , e m ∈ L∞ and Θ f d ,Θ f d ∈ L∞ As a result, s , &&m ∈ L∞ is assured based on the & boundedness of all terms on right-hand side of (35) On the other hand, taking the force filter (28) into Eq (36) yields that the force tracking error is expressed in the form: ⎛ ⎜ ⎝ % λ = ⎜1 − ⎞ % ⎟ϖ ( em , s, Θ fd , w, t ) D + η1 + k f η2 ⎟ ⎠ k fη2 where D is a differential operator Since k f η2 D+η1 + k f η (43) % is a stable filter and all signals em , s, Θ f d , w % are bounded, the bounded ϖ (⋅) implies the boundedness of λ and eλ Note that since the boundedness assumption on the fuzzy approximation error W f d is not utilized here, this proof is achieved in a global sense & Second, consider the claim (b) Since V is negative semidefinite outside the compact set { ea ea < α0 ρ w ∞ } from the inequality (42), the tracking error ea (t ) is globally uniformly ultimately bounded with convergence to a compact residual set To find the uniformly ultimate bound, we rewrite (42) as α α & V ≤ − V + ζ (t ) α1 where ζ = 2γ %f % tr(ΘTd Θ f d ) + ααρ w α1 and α = supt λmax ( M a ) > with M a = [ Λ m I n−m ]T M[ Λ m I n−m ] + [ I n−m n−m ]T Pm [ I n−m n−m ] Then, the solution of the above inequality leads to that the error trajectory of ea (t ) is shaped by 112 Frontiers in Adaptive Control ea ≤ α2 V (t0 ) exp( − 2αα01 (t − t0 )) + α2 [1 − exp( − α0 (t − t0 ))]supt ζ (t ) α1 with α = inft λmin ( M a ) > In other words, the uniform ultimate bound of ea (t ) is ea ≤ α2 supt ζ (t ) = ζ ( γ % Θ f d ∞, ρ w ∞) which can be adjusted by tuning γ and ρ Meanwhile, the residual force tracking error is adjusted by tuning η1 ,η , k f according to % lim λ (t ) ≅ t →∞ η1 % ϖ (ζ , Θ fd , w ∞ ) ∞ η1 + k f η2 (44) % with a nonnegative constant ϖ = supt ϖ (t ) dependent on ζ , Θ fd (t ) , and w(t ) ∞ ∞ Examples Components Cooperative three-link robots Mamdani SFA Mamdani FFA TS FFA Approximated term f (⋅) f d (⋅) f d (⋅) Premise Variables & q1 , q1 & q q1 d , q1 d , &&1 d & q q1 d , q1 d , &&1 d q1 d Number of fuzzy Parameters 15 ( 5× ) 32768 ( 15 ) 294912 ( 32768 × ) ( 3× ) 512 ( 29 ) 4608 ( 512 × ) ( 23 ) 720 ( × × 10 ) Approximation errors Assumedly Bounded a priori Always Bounded Always Bounded Number of Premise Variables Number of Rules Δ Δ : each premise variable has two fuzzy sets Table Comparisons between SFA and FFA Based Schemes T Third, we prove the claim (c) Consider an energy function Vs = sT Ms + em Pm em Analogous to the proof of Theorem 2, a feasible solution of the LMI (39) leads to T T T & % V s ≤ − ea Qea + s J Yd Θ fd χ − ≤ − eT Qea + a % where the fact that sT J T Yd Θ fd χ ≤ ρ2 ρ2 ρ2 χ T χ sT J T YdYdT Js + ρ2 w 2 %f % ( w + tr(ΘTd Θ fd )) χ T χ sT J T YdYdT Js + ρ2 %f % tr(ΘTd Θ f d ) has been applied Therefore, integrating both sides of the above inequality, the robust performance (40) for the motion ▓ tracking objective is assured Remark 2: The comparison between SFA and FFA based controllers applied to typical holonomic systems is made in Table From the work (Chang & Chen, 2000), the SFA-based Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems 113 & & q controller requires to take q1 , q 1, q1 d , q d, &&1 d as the premise variables In contrast, the TS FFA- based controller only needs commands q1 d as the premise variable The benefits of using the FFA-based controller (fewer rules and tuned parameters) are apparent Moreover, the fuzzy approximation error of SFA-based controllers needs to be assumedly bounded a priori ϑ13 ϑ23 ϑ22 ( x, y , ϕ ) ϑ12 ▓ y ϑ11 ϑ21 (2,0) ( 0,0) Figure A two-link planer constrained robot manipulator Simulation Example To verify the theoretical derivations, we take a cooperative two-robot system transporting an object as an application example This holonomic system is subject to a set of closed kinematic chains as illustration in Fig Two robots are identical in mass and length of links The center of mass for each link is assumed at the end of each link All the length of the first and second links l1 , l2 , and the held object are M The length of the third link is sufficiently short and is taken as a part of the object Let (x , y , ϕ ) denote the position and orientation of the held object Let ϑ1l , ϑ2 l ( l = 1, 2, 3) denote joint angles of two robots, respectively The configuration coordinate of the system is thus denoted as q1 = [ x y ϕ ]T and q = [ϑ11 ϑ12 ϑ13 ϑ21 ϑ22 ϑ23 ] Due to the fact that all the end-effectors are rigidly attached T to the common object, the holonomic constraint φ (q ) = [φ1T (q ) φ2T (q )]T ∈ R consists of ⎡ x − 0.5cos ϕ ⎤ ⎢ ⎥ φ1 (q ) = ⎢ y − 0.5sin ϕ ⎥ − ψ = ⎢ ⎥ ϕ ⎣ ⎦ ⎡ x + 0.5 cos ϕ − ⎤ ⎢ ⎥ φ2 (q ) = ⎢ y + 0.5sin ϕ ⎥ − ψ = ⎢ ⎥ ϕ +π ⎣ ⎦ 114 Frontiers in Adaptive Control ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ cos(ϑ j ) + cos(ϑ j + ϑ j )⎥ ⎥ ⎥ ψ j = sin(ϑ j ) + sin(ϑ j + ϑ j ) ⎥ , for j = 1, ⎥ ⎥ ⎥ ⎥ ⎦ ϑj1 + ϑj + ϑj T T T Therefore, the Jacobian matrix A(q ) is consists of A1 = block-diag { A11 , A12 } and A2 = block-diag { A21 , A22 } with: ⎡ ⎢ T A11 = ⎢ ⎢0.5sin ϕ ⎣ −0.5 cos ϕ ⎡ ⎢ T A12 = ⎢ ⎢ −0.5sin ϕ ⎣ 0.5 cos ϕ 0⎤ ⎥ 0⎥ 1⎥ ⎦ 0⎤ ⎥ 0⎥ 1⎥ ⎦ ⎡ − sin(ϑ j ) − sin(ϑ j 12 ) − sin(ϑ j 12 ) ⎤ ⎢ ⎥ A2 j = − ⎢ cos(ϑ j ) + cos(ϑ j 12 ) cos(ϑ j 12 ) ⎥ ⎢ 1 1⎥ ⎣ ⎦ where ϑ j 12 = ϑ j + ϑ j The kinematic transformation matrix is written as J = [ I − −( A2 A1 )T ]T In addition, the general dynamic model (21) is composed of M = block-diag { M0 , M1 , M } , C = block-diag{C , C , C } , g = [ g0 g1 g2 ]T , M0 = diag{mo , mo , I o } , C = , T g0 = [0 mo g 0] , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ a j + a j cos(ϑ j ) + a j (∗) (∗)⎥ ⎥ ⎥ Mj = a j cos(ϑ j ) + a j aj (∗)⎥ ⎥ ⎥ aj4 aj aj ⎥ ⎥ ⎦ & & ⎡ − a j sin(ϑ j )ϑ j − a j sin(ϑ j )ϑ j 12 ⎤ ⎢ a sin(ϑ ) & C j =⎢ j 0⎥ j2 ϑ j1 ⎥ ⎢ 0 0⎥ ⎣ ⎦ ⎡( a j cos(ϑ j ) + a j cos(ϑ j + ϑ j )) g/l1 ⎤ ⎢ ⎥ gj = ⎢ aj cos(ϑj +ϑj ) g/l1 ⎥ ⎢ ⎥ ⎣ ⎦ for j = 1, , where (*) represents a symmetric term; a j = (m j + m j + m j )l1 ; a j = (m j + m j )l1l2 ; a j = (m j + m j )l2 + I j ; a j = I j ; and m j , m j , m j , I j , mo , I o are system parameters The actual value of ( mo , I o , a11 , a12 , a13 , a14 , a21 , a22 , a23 , a24 ) is set as (1, 0.25, 5, 3, 3.05, 0.05, 5, 3, 3.05, 0.05) According to the holonomic constraint φ (q ) = , we Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems T can find Bg = I , τ g = [( A1 λM )T 115 T T T (τ + A2 λM )T ]T , and λg = λI , where τ = [τ τ ]T ∈ R is the applied force for the two robots; λM denotes a motion-inducing force which has T contribution to the motion of the object by A1 λM ; and λI denotes an internal force which T lies in a nontrivial null space Z = {λI ∈ R m | A1 λI = 0} Therefore, if the control input τ g is designed according to Thm 4, then the actual control input is calculated by T T τ = τ g − A2 ( A1 )+τ g T T where τ g ∈ R , τ g ∈ R are partitioned components of τ g (i.e τ g = [τ g τ g ]T ); and T T T ( A1 )+ = A1 ( A1 A1 )−1 denotes the pseudo-inverse of A1 For this cooperative two-robot system, the control objective is to track desired trajectories for the object and internal force as ⎡1 + 0.25 cos(t )⎤ ⎢ ⎥ q1 d (t ) = ⎢ + 0.25sin(t ) ⎥ , ⎢ ⎥ 0.25 ⎣ ⎦ ⎡ cos ϕ ⎤ ⎡ − cos ϕ ⎤ ⎢ ⎥ ⎢ ⎥ λgd = 40 ⎢ sin ϕ ⎥ , λgd = 40 ⎢ − sin ϕ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎣ ⎥ ⎦ where λgd and λgd represent the compressed force vector On the other hand, since the TS FFA has a general representation capability, we are able to properly choose the basis function such that fewer premise variables are used According to the function f d (⋅) , the feed-forward TS FFA-based fuzzy system (30) is constructed with &2 &2 & & q q q χ = [1 q1 d q1 d q1 d q d q d q d 1q d &&1 d &&1 d &&1 d ]T ∈ R 10 (where q1 dl is the l -th element of q1d , for l = 1, 2, 3) and linguistic variables q1dl , which accordingly are classified into two fuzzy sets From the exactly known mean and varying region, the fuzzy sets are easily characterized by the following membership functions: ⎧ μ Xnl ( q1 dl ) = − μ Xsl (q1 dl ) ⎪ ⎨ ⎪ μ Xsl ( q1 dl ) = exp( −2(q1 dl − 1) ), for l = 1, ⎩ ⎧ μ Xn ( q1 d ) = − μ X s ( q1 d ) ⎪ ⎨ ⎪ μ Xs ( q1 d ) = exp( −2(q1 d + 0.25) ), for l = ⎩ This results in the total number of fuzzy rules to be 8, i.e., Θ f d ∈ R72×10 When considering the & q special case with χ = (i.e., Mamdani FFA), all of q1dl , q 1dl and &&1dl should be utilized as linguistic variables for an admissible approximation, which needs 512 fuzzy rules and 4608 tuning parameters This implies that the proposed approach in this paper leads to less ... (1, 0. 25, 5, 3, 3. 05, 0. 05, 5, 3, 3. 05, 0. 05) According to the holonomic constraint φ (q ) = , we Global Feed-forward Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems T can find Bg... of controlling a cooperative multi-robot system transporting a common object Finally, some concluding remarks are made in Sec 100 Frontiers in Adaptive Control TS FFA-based Adaptive Fuzzy Control. .. Adaptive Fuzzy Control of Uncertain MIMO Nonlinear Systems 1 05 with symmetric positive-definite matrices G( x ) and P Similar to the proof in Thm 1, the feasibility of the LMI ( 15) and the control